**3.3. Morphologies of Lipid A phosphates**

The various morphologies of Lipid A-diphosphate and Lipid A-monophosphate nanocrystals as observed by SEM are shown in Figure 7.

**Figure 6. (A) – (C)** show overviews of HRTEM images of crystalline Lipid A-diphosphate rods at pH 7.85 (295 K) along the [111] plane. The bar in (**A)** is 50 nm, 100 nm in **(B)** and 200 nm in **(C). (D)** Shows a long rod of Lipid A-diphosphate as a SEM image, the bar is 1 m. This crystalline rod reveals FCC stacking faults along the [111] plane. A calculated Fourier transform image for models of truncated dodecahedral Lipid A-diphosphate particles is shown in **(E);** they are obtained form SAED pattern **(F)** from the lattices shown in **(A)** to (**C)**. Model calculations support a five-fold axis in the [1-10] direction, and parallel to the (001) plane, i.e. 0 and 18, but perpendicular to the fivefold axis **(E**) accounting for the experimental SAED pattern shown in **(F).**

These crystals are obtained at different particle number densities *n* as indicated and at constant T but at very low ionic strength, (*I*  10-6 M in NaCl or *I*  10-5 M in NaOH. Although the obtained nanocrystals at 10 M NaOH show a better 3d order than those grown in very low NaCl concentrations according to the corresponding SAED and SAXS pattern quality, they generally reveal SEM images of icosahedral or dodecahedral morphology. Typical sizes of these crystals are of the order of 0.3-1 m. Moreover, there was no congruent metastable other phases found for the icosahedral phase for the crystals grown in the presence of 10 M NaOH, but there was in the presence of 1.0 M NaCl, or 10 nM Ca2+ [11]. ]. Therefore, some small portion of FCC Lipid A-diphosphate was always present. Rhombic triacontahedral has been observed for Lipid A-monophosphate nanocrystals and is not the only icosahedral quasicrystal.

The antagonistic mediated incorporation into cubic crystalline Lipid A-diphosphate assemblies is more effective if the colloidal particles are oblate ellipsoids rather than curved. The assumed deformable soft spheres incorporate flexibility on a simple level [47]. At mechanical equilibrium the soft Lipid A-diphosphate sphere may be approximated into a prolate or oblate ellipsoid of revolution while preserving its volume at d3/6 where *d* is the hydrodynamic diameter of the soft sphere ( 7 nm). The distortion is given through the aspect ratio = *a*/*b* of the self-assembly, *a* is the symmetry semiaxis length of the ellipsoid and *b* is its orthogonal semiaxis length.

108 Recent Advances in Crystallography

**3.3. Morphologies of Lipid A phosphates** 

nanocrystals as observed by SEM are shown in Figure 7.

accounting for the experimental SAED pattern shown in **(F).**

not the only icosahedral quasicrystal.

The various morphologies of Lipid A-diphosphate and Lipid A-monophosphate

**Figure 6. (A) – (C)** show overviews of HRTEM images of crystalline Lipid A-diphosphate rods at pH 7.85 (295 K) along the [111] plane. The bar in (**A)** is 50 nm, 100 nm in **(B)** and 200 nm in **(C). (D)** Shows a long rod of Lipid A-diphosphate as a SEM image, the bar is 1 m. This crystalline rod reveals FCC stacking faults along the [111] plane. A calculated Fourier transform image for models of truncated dodecahedral Lipid A-diphosphate particles is shown in **(E);** they are obtained form SAED pattern **(F)** from the lattices shown in **(A)** to (**C)**. Model calculations support a five-fold axis in the [1-10] direction, and parallel to the (001) plane, i.e. 0 and 18, but perpendicular to the fivefold axis **(E**)

These crystals are obtained at different particle number densities *n* as indicated and at constant T but at very low ionic strength, (*I*  10-6 M in NaCl or *I*  10-5 M in NaOH. Although the obtained nanocrystals at 10 M NaOH show a better 3d order than those grown in very low NaCl concentrations according to the corresponding SAED and SAXS pattern quality, they generally reveal SEM images of icosahedral or dodecahedral morphology. Typical sizes of these crystals are of the order of 0.3-1 m. Moreover, there was no congruent metastable other phases found for the icosahedral phase for the crystals grown in the presence of 10 M NaOH, but there was in the presence of 1.0 M NaCl, or 10 nM Ca2+ [11]. ]. Therefore, some small portion of FCC Lipid A-diphosphate was always present. Rhombic triacontahedral has been observed for Lipid A-monophosphate nanocrystals and is

The antagonistic mediated incorporation into cubic crystalline Lipid A-diphosphate assemblies is more effective if the colloidal particles are oblate ellipsoids rather than curved. The assumed deformable soft spheres incorporate flexibility on a simple level [47]. At

**Figure 7.** SEM images of the morphologies of Lipid A-diphosphate (**A** – **C**) and Lipid Amonophosphate crystals (**D** - **E**). These crystals are usually grown from aqueous dispersions containing either 10-5 M NaCl (A, B, E) or 10 M NaOH (D, F). The icosahedral phase exhibits nanocrystals with pentagonal dodecahedral faceting. The scale bar is 100 nm in (A) - (C) and (E). The scale bar in (D) is 0.5 m. In **(F**) faceted Lipid A-monophosphate nanocrystals are shown, the scale bar is 1.0 m. The particle number density, *n*, was 350 m-3 in (A) –(C), 150 m-3 in (D) – (E), and 450 m-3 in (F).

Furthermore, some Lipid A-diphosphate and approximants were formed by a peritectic reaction from the solid Lipid A-diphosphate phase and the liquid Lipid A-diphosphate (Form C of Fig. 1) at T = 295 K. These nanocrystals possess pentagonal dodecahedral solidification morphologies, but with exclusively pentagonal faces. Since this observation is associated with molecular motion and rearrangements of the positions in the Lipid Aphosphate nanocrystals lattices above a certain temperature (T 295 K) or slowly cooling form T = 295 K to T = 288 K, which was deduced from broadening of the X-ray diffraction lines and SAED peaks, these modes belong to the phason long-wave-length phason. Relaxation of the phason strain is a diffusive process and therefore intimately related to the crystal growth process and is much slower than the phonon strain.

Quasicrystals exhibited non-crystallographic packing of non-identical Lipid A-phosphate spheres and a spatial packing of these spheres in either a cuboctahedron or an icosahedron were representative of sound physical models. Following an increase in temperature, a BCC phase was revealed and this structure gave rise to dodecagonal quasicrystals, which formed from spherical particles. Electron diffraction patterns of the dodecahedrons were recognized from the magnitudes of non-identical intensities of Lipid A-diphosphate and antagonistic Lipid A-diphosphate, which contained only 4 acyl-chains. It should be noted that the observed (3.3.4.3.4) was a crystalline analogue of the above-mentioned icosahedral quasicrystal with a different length scale. The tiling pattern of triangles (*N3*) and squares (*N4*) where the vertices were surrounded by a triangle-square-triangle tiling pattern possessed a *p4gm* plane group. Another coded Lipid A-diphosphate approximant showed an 8/3 ratio with 6-fold symmetry and plane group *p6mm*. Both (3.3.4.3.4) and dodecagonal phases revealed a *N3/N4* ratio of approx. 2.34; the ratio of the p6mm plane group was 8/3. Because bond orientational order existed, the direction of domains was classified into three orientations for the (3.3.4.3.4) tiling, but only two for the 8/3 approximants. The average magnitude of the prominent scattering vectors, ׀Q׀ = 0.121 nm-1, and the length of sides of the triangles and of the squares was 50 nm.

#### **3.4. Packing of Lipid A-phosphates**

Under the assumption that the macroscopic shape of a crystal is related to its microscopic symmetry and taking the various X-ray diffraction patterns, the SAED's and the morphology into consideration, the Lipid A-diphosphate structures and their approximants can be reconciled by lowering the symmetry from cubic, *Im* 3 *m* or *Ia* 3 *d* (*a* = 4.55 nm) for the diphosphate or monophosphate, respectively, to rhombohedral *R* 3 *m*, and finally to monoclinic (P21 or C2) which is acceptable if the Lipid A-phosphate anions were completely orientationally ordered [12]. This could be attributed to two different space-filling packings: (i) two dodecahedra on site **a** and six tetrakaidecahedra on site **d**, forming a *Pm* 3 *n* lattice; (ii) or sixteen dodecahedra on site **d** and eight hexadecahedra on site **a,** forming an *Fd* 3 *m* lattice (Fig. 8). The space-filling network of Lipid A-diphosphate consisting of slightly distorted polyhedra is reminiscent of the known basic frameworks of gas-hydrates and sodium silicon sodalites. The final geometry of the "spheres", i.e. whether they were rounded or faceted in shapes, was established. The equilibrium separation distance between the interfaces of the "spheres" suggested that they repel each other as a result of electrostatic, steric, van der Waals forces as well as water layer surrounding the spheres. The small cubic *Pm* 3 *n* (*a* = 6.35 nm) structure observed for the Lipid A-monophosphate clusters [38, 48], but different from the large cubic unit cell with *a* = 49.2 nm materialized as a result of a space-filling combination of two polyhedra, a dodecahedra and a tetrakaidodecahedra. This is in contrast to the tetrakaidecahedra (*Im* 3 *m)* or rhombodo-decadecahedra (*Fd3m*) packings observed for the Lipid A-diphosphate assemblies (Fig. 8). The small cubic *Pm* 3 *n* (*a*  = 6.35 nm) structure observed for the Lipid A-monophosphate clusters [38, 48], but different form the large cubic unit cell with *a* = 49.2 nm materialized as a result of a space-filling combination of two polyhedra, a dodecahedra and a tetrakaidodecahedra. This is in contrast to the tetrakaidecahedra (*Im* 3 *m)* or rhombodo-decadecahedra (*Fd3m*) packings observed for the Lipid A-diphosphate assemblies. **Note:** The limited number of reflections observed for this large cubic unit cell in case of the Lipid A-monophosphate was not sufficient to discriminate between primitive and body-centered-cubic lattices, or more precisely between *Pn-n* and *I- - -* extinction groups. The two differ by the reflection condition *hkl: h=k+l=2n*, which satisfied the BCC lattice, but the first set of reflections on which this condition could be tested would be the {421} reflection. This reflection was not observed because of the low resolution of the diffraction pattern.

Space groups such as *Im* 3 *m, I4* 3 *m, I4* 3 *2,* etc. are all possible and may not be separated by extinction alone. Thus, it may be argued that favorable space filling packings will be achieved when satisfactory geometrical conditions were fulfilled (Plateaus's law). For the Lipid A-phosphates this resulted from minimizing the surface area between the aqueous films and so achieving a high homogeneous curvature. However, in the case of the Lipid Amonophosphate, rhombodo-decadecahedra (*Fd* 3 *m*) packing seems to be suppressed a result of instability in the mean curvature between the tetrahedral and the octahedral nodes. Tetrakaidodecahedra packing shows only tetrahedral nodes. However, the tetrahedral angle (109.47°) can only be restored between all of the edges if the hexagonal faces of the truncated octahedron change and generate change and generate planer surfaces with no mean curvature and form Kelvin's minimal polyhedra [49,50].

**Figure 8.** Packing of *Fd* 3 *m* and *Pm* 3 *n* tetrahedrally close-packed Lipid A-diphosphate and Lipid Amonophosphate structures for two cubic crystalline phases using Wigner-Seitz cells. For both structures, aqueous bilayer compartmentalizes the hydrophobic portion of the Lipid A-phosphates into tetrahedrally networks. This network is a combination of a pentagonal dodecahedron (blue) with 14 – face polyhedra (green) in *Pm* 3 *n* and with 16-face polyhedra (green) in *Fd* 3 *m.*
